Proyectos Anal19 sis Complejo, 2016-1

Proyectos Analı́sis Complejo, 2016-1
Universidad de los Andes
16 de abril de 2016
En continuación les mando algunas propuestas que les podrı́an servir de inspiración.
1.
Meromorphic functions and singularities at infinity
La idea de este proyecto es considerar funciones meromorfas en la compactificación del plano complejo. Pueden consultar, por ejemplo, [GK06, Sec. 4.7],
[SS03].
2.
Universal analytic function
There exists a holomorphic function f such that the sequence of its derivatives
(f (n )n∈N is dense in the space of all holomorphic functions. Literatura: [BR83]
[Luh86]
3.
Infinite Products; Blaschke products
4.
Teorema de Picard
5.
Special Funtions (Gamma, Theta, Zeta . . . )
6.
Hardy spaces
Sea D = {|z| < 1} y 1 < p < ∞. Una función holomorfa f : D → C pertenece
al espacio de Hardy H p (D) si y solo si
sup
0<r<1
1
2π
Z
2π
|f (r eiϑ |p dϑ
p
< ∞.
0
Literatura: [GK06, Ch. 13.3] [Con78, Ch. 10]
7.
Hilbert spaces of holomorphic functions
Literatura: [GK06, Ch. 14].
1
8.
Diagonalization of matrices
Literatura: por ejemplo [Kat95].
9.
An operational calculus for closed operators
Literatura: por ejemplo [TL86, Ch. 5.6].
10.
Analytic Fredholm valued functions
Literatura: [She], [GGK90, XI.8]
Bibliografı́a
*Bibliografı́a
[BR83]
Charles Blair and Lee A. Rubel. A universal entire function. Amer.
Math. Monthly, 90(5):331–332, 1983.
[Con78]
John B. Conway. Functions of one complex variable, volume 11 of
Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin,
second edition, 1978.
[GGK90] Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek. Classes of linear operators. Vol. I, volume 49 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1990.
[GK06]
Robert E. Greene and Steven G. Krantz. Function theory of one complex variable, volume 40 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, third edition, 2006.
[Kat95]
Tosio Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
[Luh86]
Wolfgang Luh. On the “universality” of any holomorphic function.
Results Math., 10(1-2):130–136, 1986.
[She]
Youri Shestopalov. Introduction to the spectral theory of operatorvalued functions with applications to the solution of boundary value
problems in domains with irregular boundaries. presentation at the
mathematical seminar of the karlstad university. http://www2.kau.
se/yourshes/lecture031104.pdf.
[SS03]
Elias M. Stein and Rami Shakarchi. Complex analysis. Princeton
Lectures in Analysis, II. Princeton University Press, Princeton, NJ,
2003.
[TL86]
Angus E. Taylor and David C. Lay. Introduction to functional analysis. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, second
edition, 1986.
2